Classification of the pH-Oscillatory Hydrogen PeroxideâThiosulfate

Article pubs.acs.org/JPCA

Classification of the pH-Oscillatory Hydrogen Peroxide−Thiosulfate− Sulfite Reaction Tomás ̌ Veber, Lenka Schreiberová, and Igor Schreiber* Department of Chemical Engineering, Institute of Chemical Technology, Prague, Technická 5, 166 28 Prague 6, Czech Republic ABSTRACT: The reaction of hydrogen peroxide with thiosulfate and sulfite in acidic solution is characterized by marked temporal pH variations suggesting autocatalytic nature of hydrogen ions. When carried out in a continuous-flow stirred tank reactor this reaction provides nonlinear dynamical regimes including periodic oscillations, chaotic behavior, and multiple steady states coexisting over a range of operating conditions. The aim of the presented experimental study is a classification of the role of species and the underlying mechanism in the periodic oscillatory mode by applying single pulse additions of chosen reaction species. The external perturbations at various phases of the periodically oscillating system may cause phase advance or phase delay of the oscillations. The resulting phase transition curves are obtained for hydrogen ions, hydroxide ions, thiosulfate ions, sulfite ions, and hydrogen sulfite ions. These curves are compared with the phase transition curves calculated using the prototype mechanisms representing categories of chemical oscillators established in previous work. We found our system to be compatible with the mechanism of the category 1CX.

I. INTRODUCTION Applying perturbations on oscillatory systems and following the response puts this method of nonlinear dynamics1 to the position of a powerful tool in experimental and theoretical research. The effect of pulse perturbations can be conveniently interpreted using the phase response curves (PRC) or phase transition curves (PTC).2 The response dynamics has been studied in biological3,4 and chemical systems5−14 where spontaneous oscillations can occur. Among chemical systems, the Belousov−Zhabotinsky (BZ)15 reaction is at the forefront. Ruoff5 measured PRCs for single pulse additions of bromide ions, silver ions, and hypobromous acid to the oscillating BZ reaction mixture under closed system conditions. The author also shows that there is a good agreement between the experimental results and theoretically predicted PRCs calculated using different models based on the FKN mechanism.16 Ruoff6 extended perturbation experiments with silver ions and obtained PRCs compared with those calculated from the Explodator model and with those from his previous study. The subsequent study was focused on the phase response effects upon adding silver and bromide ions to qualitatively different oscillatory regimes using the four-variable Oregonator model.7 Dolnı ́k et al.8 examined effects of pulse additions of bromide ions to the BZ reaction in the flow-through stirred reactor. Their experimental results on response dynamics are expressed as PTCs, which were compared with the PTCs theoretically predicted for an Oregonator-type of model. In addition, the measured response curves were used to describe dynamic regimes of the periodically forced system. This work was later extended9 and partial agreement between experimental and PTCs determined from a four-variable Oregonator model was found. PRCs in the BZ reaction were also measured using additions of alkaline bromite solutions, and results of this study © 2013 American Chemical Society

were compared with the PRCs calculated using four different models derived from the FKN mechanism.10 Saigusa11 found PRCs under the condition of bubbling the reaction mixture with oxygen in the Ru(bpy)32+ catalyzed BZ reaction carried out in a closed system. Treindl et al.12 used rays of white light as pulse perturbations in the oscillating Ru(bpy)32+ catalyzed BZ reaction and studied phase resetting behavior using PRCs. Dulos and De Kepper13 obtained phase response curves for an isolated light pulse irradiation in the Briggs−Rauscher oscillatory reaction. Zagora et al.14 determined phase transition curves from experiments and from a model for the bromate− sulfite−ferrocyanide reaction perturbed by single pulse additions of hydrogen ions. The hydrogen peroxide− thiosulfate−sulfite reaction17 (HPTS) chosen for this study enters the outlined list as a new addition. The response dynamics and phase transition curves have not been previously studied in this reaction. More importantly, the use of the phase transition curves with a suitable choice of perturbing species to determine the role of those species in the mechanism and the categorization of this oscillatory reaction constitutes a new approach that helps in understanding mechanisms of oscillatory chemical reactions. The article is organized as follows: Section 2 briefly introduces the well established theory of phase resetting and the procedure for classification of chemical oscillators.18,19 In section 3, the experimental aspects of the HPTS reaction system are described including the experimental setup, operating conditions, experimental procedure for obtaining the PTCs, and the methods of numerical calculations. In Received: July 31, 2013 Revised: October 31, 2013 Published: November 1, 2013 12196

dx.doi.org/10.1021/jp407621j | J. Phys. Chem. A 2013, 117, 12196−12207

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the phase response curve. Alternatively, the phase transition curve is mapping the old phase to the new phase obtained by combining the old phase and the phase shift. Given these conditions, a graphical representation useful for experimental determination of PTC is outlined in Figure 1. The basic assumption is that the system oscillates periodically with a period T. This period is determined by monitoring the time intervals Ti = tri − tri−1 between two subsequent reference events chosen here to be the crossing of a preset pH value (reference point) when pH decreases. Steady value of Ti indicates the period T of sustained oscillations. Simultaneously, the phase is introduced as time/T(mod 1) with the origin set at tri. Certain amount of a chosen species at time tpi is injected to the system causing the perturbation. Because of the periodicity, the old phase φ is defined as (tpi − tri)/T(mod 1). The phase ranges between 0 and 1 marking the beginning and the end of the oscillatory cycle. The perturbation affects the successive oscillations causing a temporal shortening or elongation of the measured periods Ti+1, Ti+2 etc. After a sufficiently long time the system stabilizes and continues to oscillate with period T. If this transient entails more than one oscillation, the first after-transient event time tri+k can be readily used instead of tri+1 shown in Figure 1 where for the sake of clarity k = 1 is assumed. New periodic oscillations are phase shifted relative to the oscillations before the perturbation. The phase shift Δφ is obtained as (tri+1 − tr0)/ T(mod 1) and, due to nonlinearity of the system, depends on the phase of perturbation, thus Δφ = Δφ(φ). Shifts between 0 and 1/2 correspond to a phase delay, the complementary shifts refer to a phase advance. The new phase Θ may be obtained as Θ = φ − Δφ(φ) (mod 1). An alternative and from the experimentalist’s point of view, a more convenient determination of Θ is based on the time interval between tri+1 and tpi that provides the cophase coΘ (complementary phase to the new phase Θ), which is defined by Θ = 1 − coΘ. According to Figure 1, the relationship for the above-mentioned quantities is

section 4, results of perturbing the oscillations by single pulse additions of various reacting species given in terms of the phase transition curves are reported for two cases to be compared: (i) calculated PTCs for the prototype models representing relevant categories of chemical oscillators and (ii) PTCs obtained from our experiments with the HPTS reaction. Finally, in section 5, experimentally constructed and calculated phase transition curves are compared and discussed.

II. PHASE TRANSITION CURVES AND CLASSIFICATION OF CHEMICAL OSCILLATORS A. Definition of the Phase Transition Curve. Since the times of the influential book by Winfree,4 the theory underlying

Figure 1. Schematic for the experimental determination of the phase transition curve. The drawing indicates the case when the perturbation causes an immediate drop of pH and a phase delay is observed. The dashed line corresponds to hypothetical unperturbed oscillations. The pulse perturbation at time tpi corresponding to the old phase φ implies the phase shift Δφ, coΘ is the complementary new phase.

the phase resetting experiments is well-known. In geometric terms, a point at the limit cycle representing the stable periodic oscillations in the phase space is shifted off the cycle by the action of the perturbation at a particular phase and as it returns to the limit cycle a phase shift (delay or advance) of the reestablished periodic oscillations with respect to the original oscillations occurs. This phase shift can be found stroboscopically by determining times at which the returning point is indicated by a reference event. Typically, the perturbed system relaxes rapidly to the limit cycle and thus the first time indicated leads to a good approximation of the phase shift. The mapping of the perturbation (old) phase to the phase shift is known as

1 + Δφ = φ + coΘ

(1)

from where we obtain Θ either by evaluating φ − Δφ(φ) or directly from the cophase. Important relationships for the evaluation of the old phase φ, the new phase Θ, and the phase shift Δφ from experimental data are φ=

t pi − tri T

(mod 1)

(2)

Figure 2. Schematic illustration of the two types of phase transition curves. A typical form of the type 1 PTC is shown in the left panel, and the type 0 PTC is displayed in the right panel. The graph indicates domains of advanced/delayed oscillations. 12197


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tri + 1 − t pi T

(mod 1)

Θ = 1 − coΘ = 1 −

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The classification approach according to Eiswirth et al.18,19 provides two major categories of chemical oscillators, which differ from each other in their positive and negative feedback loops. The categories are distinguished according to their current cycle. A current cycle in an oscillator may be critical or strong depending on whether there is neutral or positive feedback, respectively, involving only the cyclic species. Category 1 oscillators involve a critical current cycle and contain essential species X, Y, Z, and possibly W. The presence of a strong current cycle indicates Category 2 oscillators, where the exit species is not required for the instability, and thus, just two essential species X and Z (possibly also W) are involved. The Category 1 is further divided into three subcategories, namely, 1B, 1CX, and 1CW. The notation C in these subcategories indicates that the oscillators must be operated under the continuous-flow conditions, while notation B reflects that relevant oscillators exhibit oscillations under batch conditions. For classification it is convenient to consider reduced mechanisms having essential species only. Such a reduction then leads to prototype skeleton mechanisms shown below (see refs 18, 21, and 22 for diagrammatic representation). The prototype model for the category 1B (inspired by the Oregonator model of the BZ reaction16) is as follows (□ marks a reactant in pool condition):

(3)

tri + 1 − t pi T

(mod 1)

(4)

The resulting phase transition curve for a given amplitude of perturbation is the dependence Θ = Θ(φ). Winfree 4 distinguishes two basic topological types of phase transition curves: type 1 for relatively weak perturbations and type 0 for relatively large amplitudes of perturbations (the terms type 0 and type 1 refer to the average slope of the resulting visualized curve), see Figure 2. Notice that there are two ways how the transitions between phase advance and phase delay can occur. The primary transition occurs along the primary diagonal line and corresponds to gradual deviations from in-phase oscillations, i.e., oscillations in sync. The secondary transition occurs along any of the two secondary diagonals shifted by half of the period up or down and corresponds to gradual deviations from antiphase oscillations, i.e., oscillations shifted by exactly half of the period. B. Classification of Chemical Oscillators. Generally, chemical oscillators involve a large number of species, which can be classified according to their role in the oscillations. Simultaneously, the oscillators are classified according to characteristic features of the mechanism giving rise to oscillations. First, any mechanism describing a chemical system operating at steady state can be, according to the stoichiometric network theory,20 decomposed into irreducible subnetworks called extreme currents. Oscillations emerging from the steady state via a Hopf bifurcation are a consequence of competing tendencies: one causing an instability due to positive feedback (autocatalysis) and a stabilizing tendency due to negative feedback. A dominant unstable current (typically but not necessarily an extreme one) implies autocatalysis, and the prevailing type in inorganic reactions involves a cyclic chemical pathway called the current cycle. Suitable negative feedback controlling the autocatalysis implies oscillations. In terms of participation of the species in the oscillatory behavior, the concept of essential and nonessential species was introduced by Eiswirth et al.18 An explanation of the difference between these kinds of species is well understood on the basis of their roles in the oscillatory dynamics. If the concentration of a nonessential species is kept constant (buffered) in the oscillatory regime, the oscillations can still be observed. There are three types of nonessential species. Type a is a reactant that does not interact strongly with other species, type b is an (almost) inert product, and type c is an intermediate with properties that are a combination of type a and type b species. However, maintaining constant concentration of essential species leads to suppression of oscillatory behavior. In general, four types of essential species are distinguished according to the roles they play in the dominant unstable current containing a current cycle. The autocatalytic (or cycle) species denoted as X is a part of the autocatalytic cycle. The essential species that helps to inhibit the autocatalytic process by removing a type X species is an exit species Y. The negative feedback species Z constitutes a controlling mechanism for the autocatalytic production of species X. Different paths of introducing the negative feedback loop leading to the oscillations can be recognized (tangent or exit feedback18). The fourth type of essential species that allows to reinitiate the autocatalytic cycle is the recovery species W.

k1

□ + X → 2X + 2Z k2

Z→

1 Y 2

(1B−1)

(1B−2)

k3

(1B−3)

Y→X k4

(1B−4)

X + Y→ k5

(1B−5)

2X→

In this subcategory, the unstable extreme current is formed by reactions 1B−1, 1B−2, and 1B−4, the 1-cycle in reaction 1B−1 is critical, and the current becomes unstable due to the exit reaction 1B−4. There is a negative feedback loop controlled by the species Z formed in 1B−1 and converted to the exit species Y, which helps to eliminate the autocatalytic species X when it becomes accumulated. However, the species X is not completely eliminated owing to a slow production 1B−3 so that the autocatalysis may start again. The property of internally producing all species (at the expense of a primary reactant in the pool marked by an empty square in 1B−1) gives the subcategory 1B a unique feature of oscillating under batch conditions for an extended time before the primary reactant is exhausted. Although 1B systems provide oscillatory behavior in the batch, indefinite oscillatory behavior will only occur under the continuous-flow conditions.18,23 The subcategory 1C has two variants, 1CX and 1CW. Unlike in the subcategory 1B, the species Z is consumed rather than produced by the critical current cycle. Z is typically provided by inflow even though it may also be produced internally. Its depletion provides the negative feedback mechanism leading to oscillations. The prototype mechanism of the category 1CX is k1

Z + X → 2X k2

X + Y→ 12198

(1CX−1) (1CX−2)


The Journal of Physical Chemistry A k3

Article k4

(1CX−3)

X→ k4

Y→

(1CX−4)

→ X, → Y, → Z

(1CX−5)

X→

In the 2B system, Z is produced by the autocatalytic reaction 2B−1 and simultaneously controls production of X via the exit reaction 2B−2. Thus, Z is the negative feedback species but also substitutes for the missing type Y species. This type of mechanism can be again found in some enzyme systems described by the Hill-type kinetics, an example being the comprehensive model of the complete glycolytic pathway.28 A few dozen chemical oscillators known experimentally and a number of additional abstract models were classified18,19,21 using techniques of the stoichiometric networks analysis and tools for bifurcation analysis.27,29 The classification procedure of chemical oscillators uses various methods based on experiments to distinguish between essential and nonessential species and to determine the role of the essential ones. Among these methods are quenching of the oscillations, variations of steady-state concentrations with input parameters called concentration shifts, mutual phase shifts of the oscillating species, orientation of cusp regions in the bifurcation diagram, and others.18,22,27 Here, we wish to explore yet another method based on the PTCs. Thus, we develop such a method by constructing and mutually comparing PTCs for each essential species in each category and then apply it to the HPTS reaction by comparing the PTCs of the prototypes with the experimentally obtained PTCs. A major advantage of this method is that measurement of only one species is required unlike in virtually all the previous methods.

In 1CX, the instability is due to 1CX−1 combined with the exit reaction 1CX−2, negative feedback is provided by the inflow of Z, and a slow inflow of X is strictly required to reinitiate the autocatalysis. Such mechanistic features are present in the models of several flow-controlled inorganic oscilators,18 for instance in the mixed Landolt-type reaction systems.24 As an alternative, the category 1CW is represented by the following prototype mechanism: k1

Z + X → 2X k2

X + Y → 2W k3

(1CW−1) (1CW−2) (1CW−3)

Y→X k4

(1CW−4)

Y + W→ k5

X→

(1CW−5)

→ Y, → Z

(1CW−6)

Here, the autocatalysis is initiated without direct inflow of X. Instead, the X producing step 1CW−3 is sufficient. However, the exit reaction 1CW−2 gives rise to the additional species W that subsequently reacts with Y via 1CW−4. As a result, W facilitates the recovery of the autocatalytic stage of oscillations. As with the 1CX category, there are numerous examples of inorganic 1CW oscillators.18 Moreover, the peroxidase−oxidase reaction serves as an example of a 1CW enzyme oscillator.21 The strong current cycle in Category 2 is characterized by a higher order of the autocatalytic step with respect to X than is the order of the X-removing reaction and consequently no exit reaction is needed to obtain instability. As in Category 1, there are two subcategories 2B and 2C indicating batch and continuous-flow oscillators.21 The prototype for 2C is18,19 k1

Z + 2X → 3X k2

III. EXPERIMENTAL AND COMPUTATIONAL METHODS A. HPTS Reaction System. The reaction between hydrogen peroxide, thiosulfate, and sulfite17 involves oxidation of many sulfur containing intermediates by hydrogen peroxide and is an excellent example of complex reaction networks possessing positive and negative feedback. Dynamical regimes occurring in this system depend on external constraints such as temperature, flow rate in the continuous-flow stirred tank reactor (CSTR), and presence of carbon dioxide in the reaction mixture or reagent concentrations.17 In the CSTR, the HPTS system displays interesting dynamics as the flow rate k0 is varied. The experimentally observed dynamical regimes include periodic oscillations, chaotic or other aperiodic behavior, and various stable steady states coexisting over a range of operating conditions. The reaction is also temperature sensitive.30,31 The proposed mechanism17 is derived from a related complex pHoscillatory reaction between hydrogen peroxide and thiosulfate in the presence of a catalytic amount of cupric ions.32 Complex dynamics are described by a simple eight-step reaction mechanism proposed and subsequently improved by Rábai.30,31 Main autocatalytic effect provided by the hydrogen ions catalyzed self-accelerating oxidation of hydrogen sulfite ions to sulfate was modified into two separate reactions. The mechanism is understood only partially. Its most important feature is the positive feedback in hydrogen ions largely due to the autocatalytic oxidation of hydrogen sulfite ions by hydrogen peroxide. The major inhibiting process is a rapid consumption of hydrogen ions by hydroxide ions formed in the reaction between hydrogen peroxide and thiosulfate ions. The assumed negative feedback leading to the oscillations is provided by reactions that contain hydrogen sulfite. B. Apparatus. The experiments were carried out in a cylindrical-shaped plexiglass cell holding 17.6 mL of liquid

(2C−1)

X→

(2C−2)

→ X, → Z

(2C−3)

Reactions 2C−1 and 2C−2 form an unstable current, which is controlled by Z. Category 2C systems occur naturally in heterogeneous catalysis, such as the CO2 oxidation on a single Pt crystal, where second-order autocatalysis comes from the Langmuir−Hinshelwood mechanism combined with the Pt surface reconstruction.25 It also may occur in enzyme systems described by the Hill-type kinetics that allows for a higher effective order of the autocatalytic step, as in the simplified model of the phosphofructokinase-catalyzed step in the upper glycolytic chain.26 Finally, the prototype for 2B reads21,27 k1

□ + 2X → 3X + Z k2

X + Z→ k3

Z→X

(2B−4)

(2B−1) (2B−2) (2B−3) 12199



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was measured by repeatedly applying a perturbation of the same duration (constant concentration amplitude) at different phases and registering the reference events. Throughout the experiments, the stock solutions were bubbled with nitrogen. E. Calculations. All calculations simulating the perturbations and evaluating the PTCs were done using the software package CONT.29 According to the mechanism of prototypes of oscillators for each of the relevant categories, the system of mass balance differential equations based on mass action kinetics was formulated. The corresponding rate constants and inlet concentrations of species used in calculations were taken from ref 27 where the inlet concentration−flow rate bifurcation diagrams indicate the region of regular oscillations. In each run the concentration amplitude of a species was chosen and the PTC successively constructed as follows. At first, the period of the stable periodic orbit was found from the successive intersections of the orbit with a Poincaré hyperplane at a fixed value of the species X corresponding to a reference point analogous to that used in the experiments (hydrogen ion is assumed as the type X species). The periodic orbit was incrementally perturbed at various phases assuming an instantaneous pulse and the sequence of times at the intersections of the perturbed orbit with the Poincaré section was found.

stirred by a Teflon-covered magnetic stirrer (1.3 cm long). RM6 Lauda E103 thermostat was used for keeping constant temperature of 26 ± 0.2 °C. Teflon tubes were connected to the flexible silicone tubes (ID 1.30 mm) for peristaltic feeding. An Ismatec IPC peristaltic pump was used for feeding the reactor. The external perturbations were performed at the inlet to the reactor using a syringe pump. The pH and the temperature inside the reactor were continuously measured. The pH data during the reaction were measured by a semimicro-combined pH-electrode (Theta ′90, type HC 139) connected to an Orion 525A pH-meter and an A\D converter and collected on a hard-disk of a computer (Octek, Intel Pentium 200 MHz). The computer also collected data from a temperature probe inside the reactor, controlled the instants of perturbations, and registered the reference events. C. Materials. The chemicals used in this study are H2O2 (30% aqueous solution, Penta, Chrudim), Na2SO3 (Penta, Chrudim), Na2S2O3 (Sigma−Aldrich), and H2SO4 (Lachema a.s., Neratovice). Fresh demineralized water was prepared daily and used for preparing two stock solutions. The first solution contained diluted hydrogen peroxide ([H2O2]0 = 0.0135 mol/ L) and the other contained the mixture of thiosulfate ([Na2S2O3]0 = 0.005 mol/L) with sulfite ([Na2SO3]0 = 0.0025 mol/L) and sulfuric acid ([H2SO4]0 = 5 × 10−4 mol/ L). The inlet concentration ([ ]0) for each species is the concentration of the species in a hypothetical mixed stream at the entrance to the reaction cell. The stock solutions were bubbled with nitrogen for at least 12 hours before the experiment for the elimination of carbon dioxide impurities. The values of initial concentrations for each inflow species were adopted from ref 30. D. Perturbation Experiments. The experimental procedure was initiated by introducing the prepared stock solutions to the reactor. Reaction mixture was stirred and maintained at 26 ± 0.2 °C. All experiments were carried out at the constant flow rate k0 = 0.00252 s−1. At this setting of control parameters, the system exhibits large amplitude periodic oscillations. The PTC measurements were done after the periodic oscillations were established. Because of the complexity of the measurement process, a special program33 in the graphical programming environment LabVIEW was created and used to organize perturbations and measurements. The perturbation process consists in a fast application of 0.0547 mL of concentrated solution of a perturbant to the periodically oscillating system at a chosen phase. After the perturbation, the time instants of reference events are measured until the system reestablishes the original period and then the perturbation at a new phase is applied again. To indicate the time instant of the perturbation and evaluate the period of oscillations, the reference point was chosen at pH = 6. This reference point was chosen in the range between maximum and minimum of the oscillation at the stage of the fastest decrease of pH (autocatalysis). While the amount of added solution was held constant, the concentration was systematically varied in different experimental runs. To characterize the magnitude of the perturbation, the concentration amplitude of the perturbation Δc is introduced as a control parameter. It is defined as the ratio of moles of added species and the volume of the reaction mixture; Δc is a hypothetical concentration increment in the reactor resulting from an instantaneous injection of the perturbant. The reaction was perturbed by adding hydrogen ions, hydroxide ions, thiosulfate ions, sulfite ions, hydrogen peroxide, and hydrogen sulfite ions. The phase transition curve in one experimental run

IV. RESULTS As stated earlier, our classification procedure relies on comparison of the experimental phase transition curves with those for the prototypes of the categories. Since it is the first time of attempting this approach, we need to formulate certain qualitative and quantitative characteristics derived from the PTCs. An important feature of the PTC is the shape of the curve in terms of ranges of convexity/concavity and the way they are ordered. This characteristic proved to be more suitable for the type 1 PTCs because the experimentally obtained data for the type 0 resetting often do not allow for reliable determination of details of their shape. Another useful characteristic is the order of ranges of old phases at which the perturbation causes phase advance or phase delay (corresponding to shortening or prolongation, respectively, of the affected oscillations). The third distinct characteristic is the location of intersection with the primary diagonal line, which we call the cross-point. We discretize the old phase axis into four bins centered at 0, 1/4, 1/2, and 3/4 and assign one of these values to each cross-point. The cross-point represents one way of connecting adjacent ranges of phase advance and phase delay (the other type is the crossing of the secondary diagonals, see Figure 2). Also, the cross-point corresponds to a periodic regime in a periodically pulsed system with period being an integer multiple of T. In this sense, the cross-point is stable if the slope of the PTC at this point is within the range from −1 to 1, otherwise it is unstable. We include both location and stability into our third characteristic. Additionally, there are other minor features used below to help distinguish among categories, notably the average value of the new phase in type 0 PTCs. Below we consider only Category 1 because the strong current cycle characteristic of Category 2 assumes second or higher order autocatalysis, which is not expected to occur in detailed mechanisms of inorganic homogeneous reactions based on elementary steps.18 A. PTCs Calculated for Prototypes. Figures 3−5 display calculated PTCs for various amplitudes of stimulation of each essential species of the three subcategories in Category 1. As 12200



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Figure 3. Calculated phase transition curves for the prototype of category 1B: (a) peturbation by the type X species; (b) perturbation by the type Y species; and (c) perturbation by the type Z species. Parameters: rate constants and inlet concentrations adopted from ref 27, k1 = 6.69803, k2 = 1.06601, k3 = 0.395, k4 = 63.03, k5 = 3.95, [X]0 = 0.1, [Y]0 = 0.166, [Z]0 = 1.0; the flow rate chosen within the oscillatory region, k0 = 0.6. Amplitudes Δc are shown in the box below each figure.

Figure 4. Calculated phase transition curves for the prototype of category 1CX: (a) perturbation by the type X species; (b) perturbation by the type Y species; and (c) perturbation by the type Z species. Parameters: rate constants and inlet concentrations adopted from ref 27, k1 = 0.31651, k2 = 2.43584, k3 = 0.07292, k4 = 0.05, [X]0 = 0.1, [Y]0 = 0.77, [Z]0 = 1.63; the flow rate chosen within the oscillatory region, k0 = 0.03.

described in the previous section the computational procedure differs from the experimental one only in minor details. Each graph includes five PTCs chosen within a broad range of the concentration amplitudes. The color coding of the curves in each graph is arranged according to the amplitude of the perturbation, from dark blue through light blue, green, red to

yellow. The concentration amplitudes, the rate constants, the inflow concentrations, and the flow rate are given in arbitrary self-consistent units, and their numerical values are implied by a specific choice of a reference bifurcation point in the parameter space.27 The purpose of the graphs is to obtain a representative 12201



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Figure 5. Calculated phase transition curves for the prototype of category 1CW: (a) perturbation by the type X species; (b) perturbation by the type Y species; (c) perturbation by the type Z species; and (d) perturbation by the type W species. Parameters: rate constants and inlet concentrations adopted from ref 27, k1 = 0.0083786, k2 = 0.73261, k3 = 0.025, k4 = 1.46522, k5 = 0.02, [X]0 = 0.1, [Y]0 = 3.3, [Z]0 = 7.3, [W]0 = 0.1; the flow rate chosen within the oscillatory region, k0 = 0.0048.

cross-point is slightly above 0 in 1CX, but in 1CW, the crosspoints are rather difficult to determine; upon a closer inspection, the stable cross-point is strongly shifted toward the unstable one found near φ = 1/4. At significantly higher concentration amplitudes, the PTCs switch to the type 0, which is flattening with increasing amplitude of stimulation. The range of new phase in 1CX and 1CW is mostly within [0, 0.2], while the type 0 PTC for 1B has the new phase range within [0.7, 1]. All three subcategories possess only a stable cross-point found at the same location in the type 1 PTC. The curves have always a pair of extremes: in 1CX the maximum corresponds to the maximum on the type 1 PTC, while in 1CW there is a minimum at the location of the maximum of the type 1 PTC. The PTCs obtained as a response to perturbation by the exit species Y in each subcategory (Figures 3b, 4b, and 5b) also show considerable differences. The type 1 PTCs in both 1B and 1CX undergo a concave/convex/concave sequence, but the curves in 1B start with phase delays, while those in 1CX have the opposite advance/delay pattern. Also, the region of phase delay is much larger in 1B than in 1CX. In contrast, the PTCs of type 1 in 1CW closely follow the diagonal line and have opposite convex/concave pattern. There is a pair of stable/ unstable cross-points in all three cases and their location and stability provide additional clues. The type 0 PTCs display a minimum followed by a maximum in 1CX and 1CW showing a similar range of the new phase within [0.4, 0.5] but differing

set of PTCs for comparisons with experimental phase transition curves. Within each subcategory, the PTCs of different species possess the following general features: type X and type Y species tend to have opposite characteristics of their PTCs in all subcategories; type X and type Z species have similar PTCs in 1CX and 1CW, whereas they are opposite in 1B; type X and type W species in 1CW have similar PTCs. These features are consistent with previously proposed methods for distinguishing among categories.27 Conversely, for each species, the PTCs in different subcategories differ in various aspects as described below. For the autocatalytic species X in 1B subcategory (Figure 3a) the curve of type 1 is convex within a range of low old phase values and corresponds to phase delay, and then becomes concave in a broader range of the old phase characterized by phase advance. As the concentration amplitude increases, the curvature becomes more pronounced leading to larger phase delays and advances, which is a feature shared among all subcategories. While the same pattern is found in 1CX (Figure 4a), it is opposite in 1CW (Figure 5a). The type 1 PTCs in 1B and 1CX do not develop sharp extremes, whereas in 1CW the curve exhibits a sharp maximum. Moreover, the type 1 PTCs in 1CW tend to have very small phase shifts apart from the region of phase advance near the maximum. 1B has a stable crosspoint at φ = 0 and an unstable one near φ = 1/4. The stable 12202



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Figure 6. Phase transition curves obtained from experiments with single pulse additions of H+. Amplitude of perturbations: (a) Δc(H+) = 4.3511 × 10−6 mol/L; (b) Δc(H+) = 6.2159 × 10−6 mol/L. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0 = 0.005 mol/L, [H2SO4]0 = 5 × 10−4 mol/L, k0 = 0.00252 s−1, t = 26 ± 0.2 °C.

Figure 7. Phase transition curves obtained from experiments with single pulse additions of OH−. Amplitude of perturbations: (a) Δc(OH−) = 1.2431 × 10−5 mol/L; (b) Δc(OH−) = 2.6418 × 10−5 mol/L. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0 = 0.005 mol/L, [H2SO4]0 = 5 × 10−4 mol/L, k0 = 0.00252 s−1, t = 26 ± 0.2 °C.

from the PTCs in 1B in both the order of occurrence of extremes and the range, which is within [−0.1 (mod1) ≡ 0.9, 0.1]. The shift behavior with respect to Z (Figures 3c, 4c, and 5c) provides a significant distinguishing feature between 1B and both subcategories of 1C. While the PTCs in 1C are similar to those for the species X, the PTCs in 1B display similarity to those for the species Y. There is another subtle but important feature that separates Z from X in 1CX and 1CW: while both PTCs are similar in their shape, those for the type Z species are clearly shifted to the left. B. Experimental Results. In Figures 6−10, representative experimental results are shown. The method of sequential single pulse perturbations of sustained pH-oscillations at various initial phases was used to determine the PTCs of phase transition curves. The oscillatory regime with the amplitude of pH of 2−2.5 was stimulated by pulses of constant amplitude Δc at a chosen progression of old phases φ. For each fixed Δc, this procedure was repeated to verify reproducibility. From the recorded data to each old phase φ, the new phase Θ was determined using eqs 2−4. The resulting PTCs include data from at least two measurements. Because of the expected role of the hydrogen ion as an autocatalytic species, the solution of sulfuric acid was chosen as the first perturbant. The addition causes an initial drop of pH followed by even more pronounced drop or an increase in pH depending on the phase of the perturbation. As expected, an external pulse of a very weak strength leads to the distribution of the new phases extending along the diagonal line implying

no phase shift. In successive experiments, the concentration of the injected solution was gradually increased. In Figure 6a, a well developed PTC of type 1 is shown. The perturbations near the reference point on the cycle cause phase delays, and as φ is increased, a broad range marked by phase advance occurs including a maximum of Θ. There are two cross-points. The first one located near the origin of the graph is stable, and the second one located near φ = 1/4 is unstable. The type 0 PTC shown in Figure 6b indicates mainly phase delays with one stable cross-point near φ = 3/4. Because of the rapid water dissociation equilibrium, the OH− additions is expected to play an opposite role to the H+. Thus, our second perturbant was a solution of sodium hydroxide. Each pulse was accompanied by an immediate increase in pH as a response to the perturbation followed by further increase or decrease depending on the phase of the perturbation. Starting with a low amplitude of perturbation, the type 1 PTC is obtained (Figure 7a). When compared with that for H+ (Figure 6a), the shape of this curve corroborates the expected opposite role. Initially, there is a phase advance of the oscillations followed by phase delays, both types of the phase shift are well pronounced. Likewise, the curvature sequence concave/ convex/concave is opposite. There is a maximum followed by a minimum as in the case of the type 1 PTC for acid, but their location is significantly shifted to the left. Two cross-points occur, an unstable one near the origin and another one near φ = 1/4. The latter is marginally stable (the estimated slope is roughly close to −1) due to a steeply descending part of the 12203



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Figure 9. Phase transition curves obtained from experiments with single pulse additions of SO32−. Amplitude of perturbations: (a) Δc(SO32−) = 6.2159 × 10−5 mol/L; (b) Δc(SO32−) = 1.5540 × 10−4 mol/L. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0 = 0.005 mol/L, [H2SO4]0 = 5 × 10−4 mol/L, k0 = 0.00252 s−1, t = 26 ± 0.2 °C.

Figure 8. Phase transition curves obtained from experiments with single pulse additions of HSO3−. Amplitude of perturbations: (a) Δc(HSO3−) = 4.6619 × 10−6 mol/L; (b) Δc(HSO3−) = 9.3238 × 10−6 mol/L. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0 = 0.005 mol/L, [H2SO4]0 = 5 × 10−4 mol/L, k0 = 0.00252 s−1, t = 26 ± 0.2 °C.

Figure 10 shows PTCs obtained using perturbations by thiosulfate. Like the addition of hydroxide, the perturbation causes an increase of pH. The PTC in Figure 10a is of type 1 with a concave part followed by a convex part. The curve shows mostly phase delays; some measurements indicate slight phase advance, particularly for higher values of φ, which implies two cross-points, the stable one at the origin and the unstable one near φ = 0.7. At higher concentration amplitudes the PTC is of type 0 (Figure 10b), which is marked, as most of the other type 0 PTCs but to a larger degree, by increasingly scattered data points. The left part of the curve indicates a concave part with a maximum with most of the data falling into the phase advance zone, but the right part seems to follow two branches. This nonuniqueness comes from the use of data from at least two series of measurements. Here, it seems that one series provides the upper branch with mostly phase advanced points, while another series of measurements indicates the lower branch. By giving preference to the lower branch and discarding the upper branch, the intersection with diagonal would occur near φ = 1/ 2 and would correspond to a stable cross-point. Otherwise, the cross-point would occur near φ = 3/4. However, this ambiguity does not affect the relevance of this PTC as a tool for the classification.

PTC; for lower concentration amplitudes, the PTC becomes flatter and the instability disappears. We note that the instability of this sort lead to chaotic response in the BZ reaction8 when the system was periodically perturbed. As the amount of added base is increased, switching to type 0 PTC is observed (Figure 7b). The curve is still clearly opposite of that in Figure 6b. There is one cross-point near φ = 1/4, stability of which is similar as in the case of the type 1 PTC. Experimental results shown in Figure 8 correspond to external pulses of the solution of hydrogen sulfite. Immediately after the pulse, pH dropped and the dynamics was reminiscent of the response to pulses of acid. As a result, the PTCs are quite similar to those in Figure 6. Almost all examined characteristics of both types of phase transition curve are the same, the only difference is in the location of cross-points in the type 1 PTC, where both points are shifted by about 0.15 to the left with the hydrogen sulfite as perturbant. The next perturbant is sulfite, see Figure 9. The main feature of the PTC 1 curve is a well developed pair of extremes and a tight correlation of concavity with phase advance and convexity with phase delay. The shape of this curve closely resembles the type 1 PTC for hydroxide ion additions (Figure 7a). While this may indicate similarity of the role with hydroxide in the mechanism, there are differences between type 0 PTCs (Figures 7b and 9b). The most conspicuous one is the average value of the new phase, which is about ⟨Θ⟩ = 1/2 for the sulfite and about ⟨Θ⟩ = 0 for the hydroxide. The cross-point located near φ = 0.4 is stable.

V. DISCUSSION An overview of the chosen characteristics of the calculated and experimental PTCs outlined in the previous section is provided in Tables 1 and 2. These characteristics serve as distinguishing 12204



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type X in 1B or type X/Z in 1CX. Further differentiation is subtle as none of the alternatives provides a perfect match. In category 1B both types of the PTC have the stable cross-point located directly at the origin, while in category 1CX the corresponding cross-point is shifted to the right both for X and Z. Unlike in any of the prototypes, both experimental PTCs have the stable cross-point significantly shifted to the left (i.e., to values below φ = 1). Thus, so far we cannot draw a definite conclusion and need to inspect PTCs for other species and examine how they relate to the PTC for H+. The next species producing very similar PTCs is the hydrogen sulfite ion. The PTC signatures in Table 2 are similar to those for hydrogen ions except for a distinct shift of all cross-points to the left. Such a slight shift of the entire curve is not captured by the signatures of the prototypes, but it is readily seen directly in the plots of the curves for type X and type Z species in both subcategories of 1C (Figures 4 and 5). Here, the curves for the type Z species are slightly but clearly shifted to the left relative to those for the type X. In fact, this feature is the only, even if subtle, difference between type X and type Z species in 1C. This would point to hydrogen ion being the type X species and hydrogen sulfite ion being the type Z in category 1CX, provided that we can find another clue ruling out the possibility of category 1B. One such clue is provided by the average value of the new phase ⟨Θ⟩ for the type 0 PTC. This characteristic indicates a high value (⟨Θ⟩ slightly below one) for the type 0 PTC in 1B as opposed to a low value (⟨Θ⟩ slightly above zero) for the type 0 PTC in 1C. In addition, there is a distinct minimum near φ = 1/4 on the curve in 1B, whereas at the corresponding location there is a distinct maximum for the curve in 1C. The experimental PTC for H+ (Figure 7) has both these characteristics as the type X species in 1CX. These observations lead us to the working hypothesis that the hydrogen ion is type X and that the hydrogen sulfite is type Z in the category 1CX. By using Table 2, the next two perturbants, hydroxide ions and sulfite ions, have the same signatures of the type 1 PTCs, but they differ in their type 0 PTCs. By inspecting Table 1, the role suggested by the type 1 PTCs is consistent only with type Y species in 1CX. As for the type 0 PTC, none of the species has all the signatures consistent with such an assignment. Direct examination of Figure 7b reveals that for the hydroxide ions the inconsistency stems from the PTC being significantly shifted down relative to the PTC of the prototype (Figure 4b). In other words, the average value ⟨Θ⟩ ≈ 0 for the experimental

Figure 10. Phase transition curves obtained from experiments with single pulse additions of S2O32−. Amplitude of perturbations: (a) Δc(S2O32−) = 3.1080 × 10−4 mol/L; (b) Δc(S2O32−) = 7.7699 × 10−4 mol/L. Conditions: [H2O2]0 = 0.0135 mol/L, [Na2SO3]0 = 0.0025 mol/L, [Na2S2O3]0 = 0.005 mol/L, [H2SO4]0 = 5 × 10−4 mol/L, k0 = 0.00252 s−1, t = 26 ± 0.2 °C.

signatures in the preliminary stage of identifying the role of species in the mechanism and the classification of the HPTS oscillator. If necessary, for more subtle discrimination we will examine directly the calculated and experimental curves. In pH-oscillatory reactions the role of hydrogen ion is tacitly assumed to be identical with that of an autocatalytic species X. Although this is mostly the case, we will not make this assumption a priori. Instead, we will strive to infer the appropriate role for H+ ions from the data. By comparing the signatures of the experimental PTCs of both types (Table 2) with possible matches in Table 1, the choice is narrowed to Table 1. Signatures of the Calculated Phase Transition Curves type 1 species 1B

1CX

1CW

X Y Z X Y Z X Y Z W

type 0

convex (+)

advance (+)

cross-point

convex (+)

advance (+)

cross-point

concave (−)

delay (−)

position;a stabilityb

concave (−)

delay (−)


(+) (−) (−) (+) (−) (+) (+) (−) (−) (+) (+) (−) (−) (+) (+) (−) (−) (+) (−) (+)

(+) (−) (−) (+) (−) (+) (−) (+) (−) (−)

(−) (+) (−) (+) (−) (+) (+) (−) (+) (−) (+) (−) (+) (−) (−) (+) (+) (−) (+) (−)

(+) (+) (+) (−) (+) (+)

0s 0s 0s 0+ s 0u 0+ s 1/4− 1/4− 1/4− 1/4−

s u s s

1/4 u 3/4 u 3/4 u 1/4 u 1/4 s 1/4 u 1/4+ u 3/4 s 1/4c 1/4c

(−) (+) (−) (+) (−) (+) (−) (+) (+) (−) (−) (+) (−) (+) (+) (−) (−) (+) (−) (+)

(−) (−) (−) (−) (+) (−) (−) (+) (−) (−)

(−) (+) (−) (+) (−) (+) (+) (−) (+) (−) (+) (−) (+) (−) (+) (−) (+) (−) (+) (−)

(−) (−) (+) (+) (+) (+) (+) (+)

0s 0− s 0− s 0+ s 1/2 s 0+ s 1/4 s 3/4 s 1/4 s 1/4 s

Position can be specified by appending + or − to the bin number, emphasizing that the cross-point lies to the right or left, respectively, from the center of the bin. bs, stable fixed point; u, unstable fixed point. cAdditional cross-points may occur but are difficult to distinguish. a

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Table 2. Signatures of the Measured Phase Transition Curves type 1 convex (+) concave (−) +

H OH− HSO3− SO32− S2O32−

(+) (−) (−) (+) (+) (−) (−) (+) (−) (+)

(+) (−) (+) (−) (−)

type 0

advance (+)

cross-point

delay (−)

a

(−) (+) (−) (+) (−) (−) (+) (−) (+) (−) (−) (+) (−)

position; stability −

cross-point

delay (−)


b

+

0 s 0u 1/4 u 0u 0s

advance (+) 1/4 u 1/4 s 3/4 s 1/4 s 3/4+ s

(−) (+) (+) (−) (−) (+) (−) (+) (−) (+)

(−) (+) (−) (−) (−)

3/4+ s 0+ s 3/4 s 1/2 s 1/2 s

a Position can be specified by appending + or − to the bin number, emphasizing that the cross-point lies to the right or left, respectively, from the center of the bin. bs, stable fixed point; u, unstable fixed point.

curve and ⟨Θ⟩ ≈ 0.4 for the prototype PTC. However, ⟨Θ⟩ determined from the experimental PTC for the sulfite ions is essentially the same as in the prototype PTC for a type Y species. When considering sulfite as being the type Y species, the discrepancy in Tables 1 and 2 is caused by a narrow region of phase delay near the origin due to a steep drop of the PTC, which can be considered a minor difference. Therefore, we conclude that sulfite is the type Y species in 1CX, while hydroxide ions display phase shifts inconsistent with any PTC for any essential species in the prototypes and must be assigned the role of a nonessential species. Finally, the remaining perturbant, thiosulfate ions, is also identified as a nonessential species on similar grounds. Signatures in Tables 1 and 2 indicate behavior similar to type Y or type Z species in 1B for the type 1 PTC, but for the type 0 PTC, the behavior is the same as that observed experimentally for the sulfite ions, which in turn is similar to the data calculated for the type Y species in 1CX. This inconsistency we attribute to the nonessential nature of thiosulfate. To summarize, the only consistent choice of the roles of the species and the category is that H+ is the autocatalytic species, SO32− is the exit species, and HSO3− is the negative feedback species, and the HPTS oscillator belongs to category 1CX. The remaining two species, OH− and S2O32−, cannot be consistently assigned as essential species.

nitrogen bubbled demineralized water to eliminate the dissolution of ambient carbon dioxide as well as the purity of all reaction species. Despite our efforts, the experimental noise leads to 5−10% variation of the period during a typical experiment. This observation is due to nonideal mixing as well as inherently unsteady operation and reflects itself in a visible dispersion of points making up the PTC. However, this does not preclude the use of PTCs for classification. The experimental results are used to classify this oscillatory system and to understand the role of chemical species responsible for the oscillations. For this purpose, we calculated the phase transition curves for each essential species of the prototype models for the category 1 oscillators and compared them with their experimental counterparts. We conclude that the HPTS system belongs to the category 1CX, where hydrogen ion is the autocatalytic (type X) species, hydrogen sulfite corresponds to the negative feedback (type Z) species, sulfite ion plays the role of the exit (type Y) species, and, finally, hydroxide ion and thiosulfate ion are nonessential species. Although we did not present results of our measurements with hydrogen peroxide, our observations strongly suggest that hydrogen peroxide is nonessential. In this work, we have demonstrated that the calculated phase transition curves for essential species can be used to determine the role of reaction species in an experimental oscillatory reaction system and to determine its category. It may be useful in understanding complex dynamics in other chemical systems where nonlinear oscillations occur. In a subsequent work, we will use the classification method based on the phase transition curves to test available reaction mechanisms of the hydrogen peroxide−thiosulfate−sulfite reaction and compare them with the results presented in this work.

VI. CONCLUSIONS We studied effects of external single pulse perturbations of periodic oscillatory dynamics in the hydrogen peroxide− thiosulfate−sulfite reaction system by hydrogen ions, hydroxide ions, thiosulfate ions, sulfite ions, and hydrogen sulfite ions. A brief introduction of a small volume of a dissolved reaction species elicits a transient response that eventually relaxes (typically within a few oscillations) to the original periodic attractor but with a phase shift that we interpret using the concept of the phase transition curves. During the experiments, the flow rate of inlet streams and temperature of the reaction mixture were kept constant, and the concentration of the perturbing solution was varied systematically over a wide range so that the topological type 1 and type 0 curves were obtained for all species. In addition, we attempted perturbation experiments with hydrogen peroxide, but the response was very slow and not reproducible so that the phase transition curves could not be found. This is not surprising because hydrogen peroxide is in relative surplus and hence is not expected to be an essential species (i.e., it could be buffered without losing the oscillatory dynamics). The necessary conditions for the experimental system to oscillate periodically (i.e., with nearly constant interspike periods) are properly

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AUTHOR INFORMATION

Corresponding Author

*(I.S.) E-mail: [email protected]. Tel: +420-22044165. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported from specific university research grant MSMT 21/2013, the grant GACR 203/09/2091 from the Czech Science Foundation, and the AV Č R-DAAD grant 7AMB12DE001.

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ABBREVIATIONS PTC, phase transition curve; CSTR, continuous-flow stirred tank reactor; HPTS, hydrogen peroxide−thiosulfate−sulfite reaction; BZ, Belousov−Zhabotinsky reaction 12206



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(22) Ross, J.; Schreiber, I.; Vlad, M. O. Determination of Complex Reaction Mechanisms: Analysis of Chemical, Biological and Genetic Networks; Oxford University Press: Oxford, U.K., 2006. (23) Luo, Y.; Epstein, I. R. Feedback Analysis of Mechanisms for Chemical Oscillators. Adv. Chem. Phys. 1990, 79, 269−299. (24) Edblom, E. C.; Orbán, M.; Epstein, I. R. A New Iodate Oscillator: The Landolt Reaction with Ferrocyanide in a CSTR. J. Am. Chem. Soc. 1986, 108, 2826−2830. (25) Eiswirth, M.; Bürger, J.; Strasser, P.; Ertl, G. Oscillating Langmuir−Hinshelwood Mechanisms. J. Phys. Chem. 1996, 100, 19118−19123. ́ (26) Selkov, E. E. Self-Oscillations in Glycolysis 1. A Simple Kinetic Model. Eur. J. Biochem. 1968, 4, 79−86. (27) Schreiber, I.; Ross, J. Mechanism of Oscillatory Reactions Deduced from Bifurcation Diagrams. J. Phys. Chem. A 2003, 107, 9846−9859. (28) Termonia, Y.; Ross, J. Oscillations and Control Features in Glycolysis: Numerical Analysis of a Comprehensive Model. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 2952−2956. (29) Kohout, M.; Schreiber, I.; Kubíček, M. A Computational Tool for Nonlinear Dynamical and Bifurcation Analysis of Chemical Engineering Problems. Comput. Chem. Eng. 2002, 26, 517−527. (30) Rábai, G.; Hanazaki, I. Temperature Compensation in the Oscillatory Hydrogen Peroxide−Thiosulfate−Sulfite Flow System. Chem. Commun. 1999, 19, 1965−1966. (31) Rábai, G.; Szántó, T. G.; Kovács, K. Temperature-Induced Route to Chaos in the H2O2−HSO3−−S2O32− Flow Reaction System. J. Phys. Chem. A 2008, 112, 12007−12010. (32) Kurin-Csörgei, K.; Orbán, K.; Rábai, G.; Epstein, I. R. Model for Oscillatory Reaction Between Hydrogen Peroxide and Thiosulfate Catalysed by Copper(II) Ions. J. Chem. Soc., Faraday Trans. 1996, 92, 2851−2855. (33) Bakeš, D. Study of Complex Dynamics of Chemical Reactions and Application of Dynamical Response Methods. Ph.D. Thesis, ICT Prague, 2008.

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Classification of the pH-Oscillatory Hydrogen PeroxideâThiosulfate

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